Friday, September 14, 2007

Problem 7 - Hit or get hit

Hey guys,

CAT Math is tricky and evolves every year. So here is an interesting problem for the weekend that will test your rational thinking apart from the basics of probability.

I am sure that most of you have see one of the all time best movies, "The good, the bad and the ugly" starring Clint Eastwood, Lee Van Cleef and Elli Wallach and remember the last scene of the movie where they have a 3 men duel.

Three people Good, Bad, and Ugly are playing a 'shooting at each other' game. Shooting attempts rotate cyclically Ugly first, Bad second, Good next and the order repeats. Ugly has 1/3 chance of hitting his target, Bad has 1/2 and Good has 1. A hit means that the target person gets killed and the person hit drops out.
What is the probability of survival for each of them under a rational strategy? They are also allowed to deliberately miss.

Note: Think of what is a rational strategy first

Click here for the answer


Anonymous CresceNet said...

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8:59 AM, September 14, 2007  
Blogger shubha said...

I think a rational stratergy would be where the chances of survival of all the three is maximized.

10:09 PM, September 14, 2007  
Blogger Atul said...

A rational strategy is one which minimizes the probability of being hit.
For ugly, he should deliberately miss.. as the best survival is when he misses, bad tries to hit good as he knows that if he not, good will kill him next as his probability of hit is higher than ugly. if bad kills good next ugly will try a hit at bad. otherwise he will try a hit at good after he kills bad.

Fir bad, he must try to kill good as his turn is next and his only chance of survivak is if he kills good.

For good, he must kill the person who has a high probability of shoot which is bad.

5:54 AM, September 16, 2007  
Blogger Gyan-ee said...

Shubha you are right and Atul you have taken a step further. Can you guys push a little more

4:02 AM, September 17, 2007  
Blogger Gyan-ee said...

Some basic assumptions on the rational strategy
1. All of them dont choose to deliberately miss everytime :) Else the game is not on and the probability of each surviving is 1.
2. Ugly and Bad will not shoot at each other and shoot at Good when he is alive. Because if they kill each other, then Good will kill them in the next shot
3. Good will shoot Bad when both Ugly and Bad are surviving. He wants to kill the one with higher shooting capability to increase his chances of survival

One wayy to attack this problem is conditional probability.
ugly shoots at Good and there are two conditions - Good survives or he gets killed.
Then under each scenario, there are more conditions and you go on and on and finish the problem. As you have already guessed that is the most difficult way.

Let us take an relatively easier route cutting down conditions as much as possible.
1. The first decision is to be made by Ugly - Shud he shoot or deliberately miss. We have already determined that if he shoots, he shud shoot at Good.
A) If he shoots Good and hits successfully, then both him and Bad remain in the contest with Bad getting the first shot.
Then the probability of Ugly's survival = p(Bad misses)*p(Ugly wins starting first) = 1/2*1/2 (see below for calc)
{ p(Ugly wins starting first) = p(Ugly hits bad) + p(Ugly misses)*p(Bad misses)*p(Ugly wins starting first) = 1/3 + 2/3*1/2*p(Ugly wins starting first)
Hence p(Ugly wins starting first) = 1/2 }
B) If he deliberately misses Good, then the chances of survival od Good, Bad and Ugly are as follows
p(Good survival) = p(Bad misses Good)*p(Good hits Bad)*p(Ugly misses Good) = 1/2*1*2/3 = 1/3
p(Bad survival) = p(Bad Hits Good)*p(Bad wins duel with Ugly, with Ugly starting first) = 1/2*1/2 = 1/4
p(Ugly survival) = 1-1/3-1/4 = 5/12

Since 5/12 is greater than 1/4, Ugly would choose to go with the strategy of deliberately missing the first shot. Once A misses the first shot, the probability of each survival are as given in option B
Chances of Good survival = 1/3
Chances of Bad survival = 1/4
Chances of Ugly survival = 5/12

The trick in this question is that the best has the least chance of survival. Note that the probabilities would have been the same if Bad had started. But the probabilities would change if Good had started.

Hope you enjoyed wracking the brains

12:12 AM, September 20, 2007  

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